Conventional generalized auto-calibrating partially parallel acquisitions (GRAPPA) generates uncombined coil images for each coil in an array of receive coils used by a parallel magnetic resonance imaging (pMRI) apparatus. GRAPPA reconstructs missing lines in each coil element by forming linear combinations of neighboring lines to reconstruct individual missing data points. The weights for these linear combinations are derived by forming a fit between additionally acquired lines using a pseudo-inverse operation. GRAPPA is described in Griswold, et al., Proceedings of the ISMRM, Vol. 7, Issue 6, Pg. 1202-1210 (2002).
Conventional non-Cartesian GRAPPA acquires data and makes a reconstruction kernel comprised of GRAPPA weights. The reconstruction kernel is used to reconstruct acquisition path elements acquired during a radial reconstruction. The quality of a non-Cartesian GRAPPA reconstruction depends, at least in part, on whether a suitable reconstruction kernel that corresponds to an acquisition path element being reconstructed is available. Radial GRAPPA is described in Griswold et al., Proc. ISMRM 11, 2003, p2349. While the radial trajectory provides a useful testbed for non-Cartesian trajectories, other non-Cartesian trajectories are possible. Non-Cartesian GRAPPA can include acquisition paths that include, for example, spiral acquisitions, rosette acquisitions, and so on.
An under-sampled radial acquisition will not acquire every possible ray in a radial pattern. Assuming that 360 rays are available, one for each degree in a circle associated with a radial pattern, a fully sampled data set would acquire a ray at multiple rotations (e.g., 0 degrees, 1 degree, 2 degrees). However, in an under-sampled radial acquisition, less than every ray will be acquired. For example, rays may be acquired at 0 degrees, 2 degrees, 4 degrees, and so on. Therefore there are rays missing at 1 degrees, 3 degrees, and so on. However, these missing rays can be filled in using conventional techniques to produce acceptable results. Similarly, an under-sampled spiral acquisition will not acquire every possible spiral and an under-sampled rosette acquisition will not acquire every possible rosette.
An acknowledged but tolerable error associated with conventional reconstruction assumes that a ray for 0 degrees is useful for reconstructing a neighboring ray at, for example 1 degree. Similarly, a first spiral may be useful for reconstructing a neighboring, missing spiral. This assumption holds for rays and spirals that are closely spaced as is the case when relatively mild under-sampling factors are used. However, when the two rays or two other acquisition path elements get too far apart (e.g., 0 degrees and ten degrees) this assumption and reconstruction approach no longer produces acceptable results. Also, the assumption is weaker at the edge of k-space where points are farther apart than at the center of k-space where points may intersect or nearly overlap. The review of GRAPPA provided below facilitates understanding this neighbor based approach and its shortcomings at high acceleration factors.
To better understand radial GRAPPA and the example systems and methods described below, a brief history of GRAPPA, starting at SMASH (Simultaneous Acquisition of Spatial Harmonics) is provided. FIG. 1 illustrates basic reconstruction of data in a single coil. The read-out direction is left to right. In both SMASH and VD-AUTO-SMASH a single line of data acquired in each coil in an array of receive coils in a pMRI apparatus is fit to an auto-calibration signal (ACS) line in the composite image. In VD-AUTO-SMASH this process is repeated several times and the results are averaged together to form final reconstruction weights for a reconstruction kernel that is used for reconstructing missing points. In the VD-AUTOSMASH approach, more than one ACS line is acquired in the center of k-space. This allows for multiple fits to be performed for each missing line, thereby moderating the effects of both noise and coil profile imperfections. In addition to the improved fit provided by the VD-AUTO-SMASH approach, the extra ACS lines can be included in the final image, thereby further reducing image artifacts. In both AUTO-SMASH and VD-AUTO-SMASH, as well as the original SMASH technique, the fitting process determines the weights that transform a single line acquired in each of the individual coils into a single shifted line in the composite k-space matrix. This process is shown schematically in FIG. 1. The data acquired in each coil (black circles) are fit to the ACS line in a composite image (gray circle), which in most cases is the simple sum of the ACS lines acquired in each coil.
FIG. 2 illustrates the basic GRAPPA algorithm. In GRAPPA, more than one line acquired in each of the coils in the array are fit to an ACS line acquired in a single coil of the array. In the example illustrated, four acquired lines are used to fit a single ACS line in coil number four. In GRAPPA, uncombined images are generated for each coil in the array by applying multiple block-wise reconstructions to generate the missing lines for each coil. In GRAPPA, a block is defined as a single acquired line plus the missing lines adjacent to that line as shown on the right of FIG. 2. Data acquired in each coil of the array (black circles) are fit to the ACS line (gray circles). However, data from multiple lines from all coils are used to fit an ACS line in a single coil, in this case an ACS line from coil four. This fit gives weights that can then be used to generate the missing lines from that coil. Once all of the lines are reconstructed for a particular coil, a Fourier transform can be used to generate the uncombined image for that coil. Once this process is repeated for each coil of the array, the full set of uncombined images can be obtained, which can then be combined using a normal sum of squares reconstruction.
Performing the reconstruction requires determining the weights to be used in the reconstruction. As in VD-AUTO-SMASH, a block of extra ACS lines is acquired in the center of k-space and used to determine the complex weights.
Conventional parallel imaging techniques may fill in omitted k-space lines prior to Fourier transformation by constructing a weighted combination of neighboring lines acquired by the different RF detector coils. Conventional parallel imaging techniques may also first Fourier transform an under-sampled k-space data set to produce an aliased image from each coil and then unfold the aliased signals by a linear transformation of the superimposed pixel values.
Non-Cartesian imaging has advantages over standard Cartesian imaging due to, for example, efficient k-space coverage or suppression of off-resonance effects. However, points acquired in a non-Cartesian approach do not necessarily fall onto a grid and thus have conventionally been re-sampled onto a Cartesian matrix before a Fourier transform is performed. One example gridding technique is the self-calibrating GRAPPA operator gridding (GROG) method. Using GROG, non-Cartesian MRI data is gridded using spatial information from a multichannel coil array without an additional calibration dataset. Using self-calibrating GROG, the non-Cartesian data points are shifted to nearby k-space locations using parallel imaging weight sets determined from the data points themselves. GROG employs the GRAPPA Operator, a special formulation of the general reconstruction method GRAPPA, to perform these shifts. While this re-gridding produces acceptable results in radial trajectories at low acceleration factors, at higher acceleration factors it may yield sub-optimal results.
Re-gridding has been employed in Radial GRAPPA, (Griswold, et al., “Direct Parallel Imaging Reconstruction Of Radially Sampled Data Using GRAPPA With Relative Shifts,” Proceedings of the ISMRM 11th Scientific Meeting, Toronto, 2003: 2349). Radial GRAPPA improves on conventional pMRI processing using non-Cartesian trajectories. Recall that GRAPPA determined a linear combination of individual coil data to create missing lines of k-space. GRAPPA determined the coefficients for the combination by fitting the acquired data to some over-sampled data near the center of k-space. The over-sampled data is acquired using ACS lines.
With conventional radial GRAPPA, a preliminary fully sampled scan is first performed to acquire training data that is used to estimate the missing radial data. This training data can then be used throughout a real-time scan to estimate radial lines that were not sampled. In order to calculate the required weights, multiple points in the region are used together to solve for the required number of unknown weights. Given a typical number of unknowns (e.g., 240 unknowns), a typical region size could include 8 rays and 32 points along the ray. The configuration of the different points is assumed to be the same within each region. A weight set is then derived for each region and the reconstruction is performed region by region. Note that this weight solution is the best fit solution for all of the points in the region, which is in effect correct only for the average point configuration in the region. In practical implementations, this means that some level of error is distributed to every reconstruction in the region. In addition, because only a single fully sampled data set is used for calibration, and because of its intrinsic sensitivity to variations in the point structure within each region, conventional radial GRAPPA has relied on high quality fully sampled training data that may have required extensive signal averaging. Conventionally, this acquisition may have been impractical for certain applications (e.g., contrast enhanced dynamic studies). Additionally, the errors resulting from too widely separated acquired rays has limited the maximal undersampling possible with radial GRAPPA. Similar errors may result in other non-Cartesian GRAPPA, including in spiral GRAPPA, where the distance between spiral interleaves may be so large that the regional assumption may also break down.